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Full text of "Treatise On Analysis Vol-Ii"

13 EXTENSIONS OF HERMITIAN OPERATORS 429 or unbounded) open interval in R, and 3f by the subspace &([) of indefinitely differentiable complex-valued functions on I with compact support. It is clear that the graph of D is not closed: for example, it contains the pairs (X D#) where x is of class C1 but not of class C2, and Supp(x) is compact (17.1.2). (15.13.3) Let H be a hermitian operator. Since dom(#*) is dense in E, the closure H** of H exists, and since its graph is the closure of the graph of H (15.12.4), it follows from (15.13.1.1) by continuity that H** is hermitian. We may therefore restrict our considerations to closed hermitian operators. (15.13.4) (i) Let H be a closed hermitian operator on E. For each real number a ^ 0, the closed operator H -+• ail: XH-» H • x -f odx is an injective mapping ofdom(H) into E, whose image Fa is a closed subspace ofE, and the operator (H + a/7)"1 (15.12.5) is continuous on Fa. (ii) The linear mapping (15.13.4.1) V : x^H-iTKH+iir1 ' x of Pi into E is an isometry ofV^ onto the closed subspace V(F1) = F_A. The mapping I — V: F1 = dom(F) -+Eisa bijection 0/dom(F) onto dom(H), and we have (15.13.4.2) H-y = i(I+ V)(I- V)~l 'y for all y e dom(/f). (iii) Conversely, let F be a closed subspace ofE, and U an isometry off onto a subspace o/E, such that the image GofF = dom(U) under /— U is dense in E. Then the operator I — U is a bijection ofF onto G; if, for each y e G, we put (15.13.4.3) H-y = i(I+ U)(I- U) then H is a closed hermitian operator, with dom(H) = G; moreover, the operator V defined in (15.13.4.1) is equal to U. By virtue of (15.13.1.1) we have (15.13.4.4) \\H-ynging to ^